Fabricated shape estimation for droplet based additive manufacturing

ABSTRACT

A geometry of a substrate surface is received at a neural network. The neural network is trained using one or more training sets. Each training set comprises a different type of substrate geometry and a collection of manufacturing process parameters. The substrate is configured to receive at least one liquid droplet. A shape of the at least one droplet after it has been deposited on the substrate is determined based on the received geometry. An output representing the determined shape of the at least one droplet is produced.

TECHNICAL FIELD

The present disclosure is directed to the design of mechanical parts.

BACKGROUND

Recent advances in additive manufacturing technologies have triggeredthe development of powerful design methodologies allowing designers tocreate highly complex functional parts.

SUMMARY

Embodiments described herein involve a method comprising receiving ageometry of a substrate surface at a neural network. The neural networkis trained using one or more training sets. Each training set comprisesa different type of substrate geometry and a collection of manufacturingprocess parameters. The substrate is configured to receive at least oneliquid droplet. A shape of the at least one droplet after it has beendeposited on the substrate is determined based on the received geometry.An output representing the determined shape of the at least one dropletis produced. Embodiments involve a method comprising receiving one ormore training sets and a collection of manufacturing process parametersat a neural network and a collection of manufacturing process. Eachtraining set comprises a different type of substrate surface geometry.The neural network is trained using the one or more training sets. Ageometry of a substrate is received at the neural network. The substrateis configured to receive at least one liquid droplet. A shape of the atleast one droplet after it has been deposited on the substrate isdetermined based on the received geometry. An output representing thedetermined shape of the at least one droplet is produced

Embodiments involve a system comprising a processor. The systemcomprises a memory storing computer program instructions which whenexecuted by the processor cause the processor to perform operations. Theoperations comprise receiving a geometry of a substrate surface at aneural network. The neural network is trained using one or more trainingsets. Each training set comprises a different type of substrate geometryand a collection of manufacturing process parameters. The substrate isconfigured to receive at least one liquid droplet. A shape of the atleast one droplet after it has been deposited on the substrate isdetermined based on the received geometry. An output representing thedetermined shape of the at least one droplet is produced.

The above summary is not intended to describe each embodiment or everyimplementation. A more complete understanding will become apparent andappreciated by referring to the following detailed description andclaims in conjunction with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a process for predicting the shape of a solidified dropleton a substrate in accordance with embodiments described herein; FIG. 2shows an output simulation having the substrate and the droplet inaccordance with embodiments described herein;

FIGS. 3A, 3B, and 3C show examples of substrate geometriesrepresentative of the three training sets in accordance with embodimentsdescribed herein;

FIGS. 4A-4F show example representative outputs from the three trainingsets in accordance with embodiments described herein;

FIGS. 5A and 5B show an examples of an estimated height fields of ashape in accordance with embodiments described herein;

FIG. 6A shows an example drop-on-demand magnetohydrodynamic (MHD)deposition system in accordance with embodiments described herein;

FIG. 6B shows a simulation model showing the magnetic field generated bythe pulsed magnetic coil as well as the volume fraction of ejectedliquid aluminum in accordance with embodiments described herein;

FIG. 7 shows the solid fraction of the droplet on the substrate surfaceat different times in accordance with embodiments described herein;

FIG. 8 illustrates a temperature map of the solidifying droplet thatshows a gradual cooling of the droplet and heat transfer to thesubstrate at different times in accordance with embodiments describedherein;

FIG. 9 illustrates a temperature map of two droplets in accordance withembodiments described herein;

FIG. 10 shows the first four Laplacian eigenfunctions on a square domainwith homogeneous Dirichlet boundary conditions visualized as a surfacein 3d in accordance with embodiments described herein;

FIG. 11 shows an image of the top view of a set of lines printed by a 3dprinter in accordance with embodiments described herein;

FIG. 12A illustrates an 2D array output of the height field inaccordance with embodiments described herein;

FIG. 12B shows a voxelized representation of a single droplet inaccordance with embodiments described herein;

FIG. 13 illustrates a volumetric plot of the voxels probabilities wherethe color intensity corresponds to the probability in accordance withembodiments described herein;

FIG. 14 shows a surrogate model for computations using a neural net inaccordance with embodiments described herein;

FIG. 15 shows a plot of time versus the number of droplets in accordancewith embodiments described herein;

FIGS. 16A-16C illustrate representations of the as-manufactured shapesin accordance with embodiments described herein; and

FIG. 17 shows a block diagram of a system capable of implementingembodiments described herein.

The figures are not necessarily to scale. Like numbers used in thefigures refer to like components. However, it will be understood thatthe use of a number to refer to a component in a given figure is notintended to limit the component in another figure labeled with the samenumber. (2)

DETAILED DESCRIPTION

The present disclosure relates to additive manufacturing (AM). Many suchmethods often operate under the assumption that any designed shape canbe fabricated, e.g., using a 3D printer, and the resulting part matchesthe designed shape perfectly or with negligible errors. In reality,however, the selected AM process, mechanical characteristics of theprinter or the material being used may introduce significant limitationsto printability of a particular design in the form of a minimumprintable feature size.

Embodiments described herein involve a computationally efficientalgorithm for the estimation of as-printed shape in a droplet-basedadditive manufacturing process.

Uncertainty in the manufacturing process may lead to a range of possibleas-printed shapes. The present disclosure first describes thecomputation of the perfect as-printed shape, i.e. where the additivemanufacturing machine exactly executes a given plan, and also describesthe prediction of the as-printed shape when accounting for manufacturingimperfections such as variability in droplet deposition location,frequency, temperature, and other parameters. Prediction of theas-printed shape involves the solution of highly non-linear timedependent system of partial differential equations capturing complexphysics such as multiphase flow and heat transfer in three dimensions.Solving such a complex system of equations on a high-end modern-daydesktop computer is computationally expensive. For instance, simulatingthe droplet deposition physics of one droplet using a high-fidelitymodel involves approximately an hour of computer time. In order tosimulate the building of a part, one may involve millions (or evenbillions) of droplets. Hence, to deal with such complexity, embodimentsdescribed herein involve an efficient reduced-order model for liquiddroplet deposition physics based on hybrid machine learning. Thisreduced order model has an advantage of decreasing the computationaltime for simulating droplet deposition to 1 millisecond. Using thishighly efficient reduced-order model enables estimation of the shape ofthe printed part in minutes, thereby enabling a user to visualize theresult of a printing process plan without actually running the printer.

Embodiments described herein show a way to predict the shape of thesolidified droplet under various conditions. One possible solution is tosolve a high-fidelity model such as the thermo-fluidic multiphase modelsuitably tracking the solid-liquid interface. One simple way to modelthe solidification is by adding an extra temperature dependent momentumloss term to the Navier-Stokes equation. The drawback of such a highfidelity model is its high computational cost. In fact, a single dropletdeposition simulation (using such models) can take more than an hour ona high-end modern desktop computer. Thus, computer simulation ofmillions of droplets to predict the part shape is prohibitivelyexpensive and hence not feasible. According to various embodiments, onecould develop an analytical solution for the shape of the solidifieddroplet on a flat surface to predict the shape of the solidified dropleton a flat surface. These methods do not account for the effect of thesubstrate geometry. This is a limiting factor as the substrate geometryis found to have a high impact on the shape of the solidified droplet,and resulting mechanical properties. FIG. 1 shows a process forpredicting the shape of a solidified droplet on a substrate inaccordance with embodiments described herein. A geometry of a substrateis received 110 at a neural network. The substrate is configured toreceive at least one liquid droplet. The neural network may be trainedusing one or more training sets, each training set comprising adifferent type of substrate geometry, and a collection of manufacturingprocess parameters. According to various implementations, the trainingsets comprise one or more of a slightly curved surface, a highly curvedsurface, a rough surface, and a step-like surface. The received geometryof the substrate may be a 3D representation of the substrate.

A shape of the at least one droplet is determined 120 after it has beendeposited on the substrate based on the received geometry. According tovarious embodiments, the shape comprises a 3D representation of the atleast one droplet after it has deposited on the substrate. According tosome implementations, determining a shape of the at least one dropletafter it has been deposited on the substrate comprises determining theshape of the at least one droplet using one or more of a high-fidelitymodel and a steady-state model.

An output representing the determined shape of the at least one dropletis produced 130. The output may be a 3D representation of the shapeobtained by determining the shape based on the received geometry. Theshape of the as-printed part can be estimated using convolution of aprobability density function of all possible droplet shapes with thetool path. This method also enables efficient estimation of partgeometry. However, it is only a probability field describing the partgeometry and does not capture the droplet level (local) influence on theas-printed part (global) geometry. According to embodiments describedherein, a shape of a product part is estimated based on the output wherethe product part comprises a plurality of droplets. High fidelityembodiments described herein have very high computational efficiencycompared to more traditional approaches. This efficiency is achieved byreducing the model complexity using a hybrid machine learning approach.The core idea of this method is to train an Artificial Neural Network(ANN) to learn the steady-state shape of a liquid droplet falling on asubstrate of arbitrary geometry. Specifically, the input to the networkis the substrate geometry and the output from the network is the shapeof the solidified droplet. Once the network is trained (which is aone-time effort), predicting the shape of the solidified droplet for anygiven geometry boils down to a set of matrix-vectormultiplications/additions. These basic linear algebra operations can beperformed very efficiently via parallelization (on the CPU and/or GPUcores). Thus, a trained network can predict the shape of the solidifieddroplet very efficiently. Another advantage of this approach is that thetraining data set can be obtained via a combination of high-fidelitymodel (such as thermo-fluidic multiphase flow simulation), simplifiedmodel (such as the steady-state shape of a liquid droplet) and/or actualexperiments.

According to various embodiments, the training data set is generatedbased on a simplified steady state simulation of liquid dropletdeposition on randomly generated substrate geometry. Although thesteady-state simulation is computationally efficient, they do notcapture all the relevant physics at the droplet-scale. Hence, to improvethe accuracy of the steady state simulations, the training set may beaugmented with the shape of the solidified droplet from directexperimental measurements. In some implementations, the steady-statemodel can be combined with the high-fidelity solution to furtheraccelerate the generation of training data set.

The computational efficiency of embodiments described herein may beachieved by using a series of approximations and model order reductions.The dimension of the 3D droplet shape may be reduced to a 2D heightfield. It may be assumed that a solidified droplet does not change itsshape. The latter assumption allows decoupling of the deposition eventsand thereby reduce the problem to the deposition of a single droplet onsome curved surface. In general, the decoupling is not valid as thesolidified droplet geometry depends on droplet deposition speed and rateof heat transfer from neighboring droplets. In other words, thedecoupling is valid only when the frequency of droplet deposition is lowand/or when the droplets are deposited in a non-overlapping fashion. Thehigher the frequency of the droplet deposition, higher is the heat fluxand thereby inducing re-melting of solidified droplets. Dropletdeposition in a nonoverlapping manner avoids re-melting as it avoidsdirect droplet interaction and thus local heat transfer. Thirdly,Artificial Neural Networks (ANN) are used to reduce the computationalcomplexity of the droplet model. The neural network has two maincomponents. The first component is the design of network architecturei.e. number/type of layers/neurons and its interaction. The second maincomponent is the training data set employed to train the neural network.

For an embodiment of the network structure, a combination of linearlayers, hyperbolic tangent activation functions, convolution layers, andbatch normalization layers. The final layer has ramp activationfunctions to suitably threshold the output. As part of this approach,the training set is created that includes the shape of the substrate asinput and the height field of the solidified droplet as output. Here,the height is measured as the distance from the deposition substrate toa camera or sensor mounted above the substrate. To create the trainingset, software may be used to predict the steady-state shape of thedroplet by minimizing the total droplet energy. The droplet energy maybe computed by accounting for the effect of surface tension, pressure,and/or gravity. An output simulation having the substrate 220 and thedroplet 210 is shown in FIG. 2.

According to embodiments described herein, two free adjustableparameters: namely gravitational constant and droplet contact angle areused. By adjusting these parameters, it can be ensured that the steadystate shape of the droplet on the flat surface matches the shapeobserved via real experiments. Training an ANN may involve using a largerepresentative training data set. Here, three sets of training data setsrepresenting various types of substrate geometry were generated. Eachtraining set has about 10,000 data points. Examples of the droplet shape310, 330, 350 from each training set are shown in FIGS. 3A-3C. The threetraining data sets differ primarily in the nature of the substrategeometry. FIGS. 3A, 3B, and 3C show examples of the first set, secondset, and third set, respectively. In the first set, the substrategeometry 320 is a slightly curved surface. In the second set, thesubstrate geometry 340 includes a step-like surface having one or morestep features 345. The third set includes substrate geometries 360having highly curved surfaces.

To generate a random surface for the first set, the substrate geometrycan be represented as a linear combination of harmonic basis functions,among other techniques to represent curved surfaces such as splinesurfaces with control points, patched surfaces, subdivision surfacesetc. For every such surface, the steady-state droplet shape iscalculated. The height field of the deposition surface is then exportedin addition to the droplet height as measured from the depositionsurface (i.e. offset from the deposition surface) into files. Thegeometry of the deposition surface is the input to the network anddroplet height (i.e. offset) is the output from the network. Each inputand output is an image, for example.

The examples from the training set are shown in FIGS. 4A-4F. FIG. 4Ashows an example substrate geometry represented as a height field. Inthis example, first sections 410 are further away than second sections420. While two different sections are shown, it is to be understood thatthere is a spectrum of heights represented in the height field. FIGS.4B-4D show example solidified droplets. Similarly, FIG. 4E illustratesanother example substrate geometry represented as a height field. Inthis example, First sections 450 are further away than second sections460. FIG. 4F shows an example solidified droplet.

After training the neural network, the printing of a part can beestimated by iteratively placing droplets along the path (as determinedby the G-Code) slice by slice. Therefore, an output representing anestimated shape is produced. FIGS. 5A and 5B show an examples of anestimated height fields of a shape in accordance with embodimentsdescribed herein.

Uncertainties in any manufacturing process lead to deviations betweennominal designs and their fabricated counterparts. Nominally designedshape and material layout is invariably altered in an AM process, andshape variations can lead to undesirable effects such as (unintentional)porosity and surface roughness. These in turn can lead to long-termperformance degradation due to residual stresses, fatigue failuremechanisms such as crack initiation, or can negatively affect bulkmechanical properties. Metal parts designed for high stress applicationsshould be fully dense with smooth surfaces to minimize the possibilityof failure in service. While such properties are achievable inmachining, the geometric complexity achievable using AM enablesmanufacturing functional high performance lightweight parts that may beimpossible to fabricate otherwise. This feature of metal AM motivatesthe desire to understand the relationship between a nominal design andits corresponding variational class of shapes arising due to thecombination of chosen AM process parameters and manufacturing error.Although embodiments described herein are directed to select metal AMprocesses, it can be observed that similar issues arise in polymer AM aswell.

The intricate relationship between AM process parameters and fabricatedpart properties has received significant attention, mostly byfabricating parts and either studying their microstructure or bymechanical testing to determine (anisotropic) material properties. Formetal AM, microstructural details such as grain morphology, graintexture, and phase identification for Powder Bed Fusion and DirectEnergy Deposition processes are studied using LOM and SEM microstructureimaging. The orientation of the columnar grains seen in these processesare highly influenced by a combination of the scan strategy and appliedenergy to induce material phase changes key to the AM process.Experimental analysis to map process parameters to particularmanufacturing-driven structural and material variation is done in acase-by-case manner for each material and process combination in metalAM processes. Due to the availability of several AM technologies,applications, and testing strategies, a rich body of literature existsfor AM metallurgy and processing science.

AM is not a stand-alone process and is typically followed by heattreatment to relieve residual stress and/or improve mechanicalperformance, and by machining to improve part surface quality and/orremove support materials. Applications using post-processing may planthe post-processing operations, such as support material removal orfinishing rough part surfaces. Equipment manufacturers are concernedwith geometric properties such as the minimum feature size/resolution,surface roughness, and accuracy to ensure the overall fabricated shape(excluding support materials) is as close as possible to the nominaldesign. Therefore, estimating the fabricated shape corresponding to anominal design may be done as a function of AM process parameters, sothat important properties such as porosity, roughness, and geometricdeviation from nominal design can be characterized before fabrication.Very little attention has been directed towards computational modelingand representation of as-manufactured part shape. Effective shapemodeling of as-manufactured part shape will help AM process planning byeliminating expensive trial-and-error due to multiple builds, andconverge quickly to parameter values that yield acceptable part quality.

Embodiments described herein illustrate an approach to efficientlyestimate as manufactured shape as a function of AM process parameterswhile considering manufacturing uncertainty. The approach uses classicalideas of predictive estimation with uncertainty quantification.Experimental data is first assimilated in the form of scanned prints ofsimple shapes for a fixed set of process parameters. The inputuncertainties associated with local material deposition are quantified.This is done by solving an inverse problem to estimate parameters of aprobability kernel whose convolution with the tool-path used to buildthe scanned parts yields the best representation of the family ofscanned shapes. This inverse problem is solved in two stages. First thefamily of scanned shapes are mapped to a unparameterized representationof the kernel by solving an iterative deconvolution algorithm. A goodinitial condition for the iterative algorithm is obtained by solving amulti-physics problem that captures the AM physical phenomena at thesmallest manufacturing scale (i.e. the scale of the minimum featuresize). The unparameterized field resulting from the deconvolutionapproximates the spatial probability of material deposition (coupledwith measurement error) for the smallest manufacturable feature. Thisfield is then parameterized in terms of a known spatially varyingfunction (e.g. a Gaussian distribution) considered as the probabilitykernel. Solving for the kernel parameters that best approximate theunparameterized field results in a mapping from the fixed AM processparameters to the kernel parameters. Repeating this process for severalexperiments by varying the process parameters produces a data-set thatis fed into a neural network that learns the mapping from AM processparameters to kernel parameters. Thus, given a set of AM processparameters the kernel parameters that capture the uncertainty in localmaterial deposition can rapidly be estimated. Embodiments describedherein may use a drop-on-demand magnetohydrodynamic (MHD) depositionsystem shown in FIG. 6A to illustrate the computational approach. It isto be understood that other types of printing processes and/or systemsmay be used.

In the MHD method, a spooled solid metal wire (e.g., aluminum wire) 610is fed continuously into a ceramic heating chamber 620 of amagnetohydrodynamic printhead and resistively melted to form a reservoirof liquid metal that feeds an ejection chamber via a capillary force. Acoil 625 at least partially surrounds the ejection chamber and iselectrically pulsed 640 to produce a transient magnetic field thatpermeates the liquid metal and induces a closed loop transient electricfield within it. The electric field gives rise to a circulating currentdensity that back-couples to the transient magnetic field and creates amagnetohydrodynamic Lorentz force density within the chamber. The radialcomponent of this force creates a pressure that acts to eject a liquidmetal droplet out of the nozzle orifice. Ejected droplets travel to asubstrate 650 where they coalesce and solidify to form extended solidstructures. Three-dimensional structures are printed layer-by-layerusing a moving substrate 650 controlled by a controller that enablesprecise pattern deposition. FIG. 6B shows a simulation model showing themagnetic field generated by the pulsed magnetic coil as well as thevolume fraction of ejected liquid aluminum.

Embodiments described herein involve a way to construct theas-manufactured part shape by solving the forward problem of tracing thematerial accumulation along a specified deposition pattern (ortool-path). Solving this well-posed problem involves a local estimationof the material accumulated at each location along the depositionpattern. Therefore, the physics of local material accumulation is besolved at the smallest manufacturing scale, i.e. the scale of theminimum printable feature. For the drop-on-demand MHD system, estimationof local material accumulation involves modeling the multi-physicsproblem that captures droplet coalescence. The minimum printable featureis a solidified droplet whose shape is dependent on a combination ofprocess parameters, namely the droplet temperature, depositionfrequency, the shape of the substrate on which deposition occurs (aslayers build up, surface roughness and curvature will influencecoalescence), the deposition pattern, and manufacturing uncertainty.

A process for estimating an as manufactured shape while consideringmanufacturing uncertainty is described in accordance with embodimentsdescribed herein. A plurality of scanned prints of a product part and ascan path is received.

According to various implementations, the part is printed using amagnetohydrodynamic deposition system. According to embodimentsdescribed herein, the part comprises a plurality of minimum printablefeatures deposited along the scan path. A shape of a minimum printablefeature of the product part is determined by analyzing the respectiveprints in a scan path representation. In some cases, the shape of theminimum printable feature is determined by statistically analyzing therespective prints in the scan path representation. According to variousembodiments, the minimum printable feature comprises a droplet after ithas solidified on a substrate. The shape of the minimum printablefeature may comprise a height of the minimum printable feature after ithas solidified on a substrate.

A manufacturing error of the minimum printable feature, that isdependent on the combination of process parameters including thesubstrate shape as discussed above, is determined based on thestatistical analysis. According to various implementations, determiningthe manufacturing error of the shape of the minimum printable featurecomprises determining the manufacturing error of the shape of theminimal printable feature using estimated shapes of the minimumprintable features on a plurality of randomly generated surfaces. Insome cases, determining the manufacturing error of the minimum printablefeature comprises determining the manufacturing error of the minimumprintable feature using a multivariate Gaussian. A manufacturing errorof a shape of the part is determined based on the determinedmanufacturing error of the minimum printable feature. According tovarious implementations, the manufacturing error of the shape of thepart comprises sweeping the minimum printable feature with the scanpath. According to various embodiments described herein determiningmanufacturing error of the shape of a part, and the estimatedmanufactured shape comprises assuming that the plurality of minimumprintable features the same shape at every location. An estimatedmanufactured shape of the part is produced based on the determinedmanufacturing error of the part.

According to various embodiments, first the forward problem is solvedwithout considering manufacturing uncertainty and present amulti-physics model for droplet coalescence on curved surfaces. Thesolution to this multi-physics problem may involve significantcomputational resources and will be impractical to solve on adrop-by-drop basis for practical parts manufacturing. However, solvingthis numerical problem offline for several combinations of processparameter values helps build a training set that can be used toconstruct a surrogate model for the discretized PDE solver, whileconsidering manufacturing uncertainty. For the drop-on-demand MHDsystem, local material accumulation on a surface with Gaussian curvatureκ is modeled by solving the system of coupled multiphase incompressibleNavier-Stokes and heat transfer equations,

$\begin{matrix}{{{{\nabla{\cdot u_{i}}} = 0},{{\frac{\partial u_{i}}{\partial r} + {\left( {u_{i} \cdot \nabla} \right)u_{i}} - {v_{i}{\nabla^{2}u_{i}}}} = {{- {\nabla p}} + g + f_{\sigma} + {{D(T)}u_{i}}}},{f_{\sigma} = {\sigma\kappa{\nabla\alpha}}}}{\frac{\partial u_{i}}{\partial t} = {\Delta T_{i}}}} & (1)\end{matrix}$

Here, the subscript i denotes the phase, for example, liquid or gaseous,t is the time, u_(i) is the velocity vector, v_(i) is the kinematicviscosity, p is the pressure, g gravitation constant, is the force whichmodels surface tension, D(T) is Darcy's term which models the phasetransition, σ is the surface tension constant, κ is the curvature of theboundary between phases and T_(i) is the temperature. Time steps of 1 msare used to obtain the intermediate droplet solidification andtemperature profiles. FIG. 7 shows the solid fraction of the droplet onthe substrate surface at different times. Time step 710 shows thecompletely liquid droplet before it is on the substrate. Time steps 712,714, 716, 718, 720, 722, 724, 726 show the droplet at various stages ofsolidifying and time step 728 shows the droplet completely solidified onthe substrate. Similarly, FIG. 8 illustrates a temperature map of thesolidifying droplet that shows a gradual cooling of the droplet and heattransfer to the substrate at different times. Time step 810 shows thecompletely liquid droplet before it is on the substrate. Time steps 812,814, 816, 818, 820, 822, 824, 826 show the droplet at various stages ofsolidifying and time step 828 shows the droplet completely solidified onthe substrate.

While FIGS. 7 and 8 show simulations are conducted for the case of asingle droplet falling on a flat surface, it is to be understood thatembodiments described herein could be used to determine a solidifiedshape of multiple droplets on a surface. When multiple droplets aredeposited and simultaneously solidifying, the curvature and temperatureof previously deposited and solidifying droplets affect theas-manufactured shape. Droplet coalescence is shown in FIG. 9 where thecoupled evolving temperature profile can be seen. At time step 910, afirst droplet 922 and a second droplet 924 have yet to reach thesubstrate 920. At time step 930, the first droplet 922 has made contactwith the substrate 920 and has started to cool and solidify whiledroplet 924 has not reached the substrate yet. At time step 950, boththe first droplet 922 and the second droplet 924 have reached thesurface of the substrate 920 are starting to solidify. Finally, timestep 970 shows a later time step than that of 950 with both the firstdroplet 922 and the second droplet 924 have reached the surface of thesubstrate 920 are starting to solidify. It can be observed that thefirst droplet 922 is further along in the solidification process becauseit was deposited before the second droplet 924. Each time stepsimulation takes around one hour to complete. Thus, the simulation ofas-manufactured part shape using fully coupled numerical methods willnot scale to realistic parts.

Embodiments described herein involve a way to characterize theuncertainty in the manufacturing process at the smallest manufacturingscale (i.e. the droplet scale). To do this, a training set is built byrunning simulations for droplet deposition on a randomly generatedsurface with varying Gaussian curvature. It is important to capture thedroplet solidification on curved surfaces because the final part surfaceroughness and porosity is directly influenced by the accumulated buildupof material on previously solidified layers (which cannot be consideredas a flat substrate). Embodiments described herein involve generating arandom surface using a linear combination of six basis functions, whereeach basis function is an eigenfunction of the (2D) Laplace operator ina square domain with homogeneous Dirichlet boundary conditions, and thelinear coefficients are random values. The Laplacian eigenfunctions inthis simple case are trigonometric function of two variables. FIG. 10shows the first four 1010, 1020, 1030, 1040 Laplacian eigenfunctions ona square domain with homogeneous Dirichlet boundary conditionsvisualized as a surface in 3d. Weighted sums of the eigenfunctions arerepresented as surfaces embedded in R3. Many other approaches togenerate random curved surfaces are possible.

In addition to the substrate Gaussian curvature κ at the depositionpoint, the droplet temperature, Td and the substrate temperature, Ts areconsidered as the key process parameters driving the droplet shape. Thesolidified droplet shape can be non-trivial, and simulation times cantake around one hour of compute time for grid sizes of 1283. The outputof each simulation at the final time step is the solidified dropletrepresented as a binary-valued (indicator function) on athree-dimensional grid, but it is highly unlikely that the printeddroplet will have exactly the same shape as the simulated droplet. Thisis due to manufacturing and material accumulation error. To model thiserror, several droplets are printed at distinct (nonoverlapping)locations for fixed values of T_(d), T_(s), κ using the MHD process.These dissimilar shapes are individually scanned using a Kaysen surfacescanner that has a scanning resolution of 20 μm in each spatialdirection. FIG. 11 shows an image of the top view of a set of linesprinted by a 3d printer.

The image is acquired by the Kaysen scanner. The color coding indicatesthe height of the scanned pixel relative to the camera position. FIG.12A shows a similar image as FIG. 11, but shows a close-up at one of thepaths. The output from the scanner is a 2D array of the height field asshown in FIG. 12A, which is an image storing the height values betweenthe scanner position and points on the printed part at sample points inthe scanning plane. Assuming the droplet shape does not have undercuts,this height map is sufficient to reconstruct a 3d representation of themanufactured droplet P_(m)(T_(d), T_(s), κ) as an indicator function. Anexample of the voxelized representation of a single droplet is shown inFIG. 12B. Finally, the probability that a voxel is filled with thematerial is calculated by taking the average value of all P^(m)(T_(d),T_(s), κ). The resulting volumetric plot of the voxels probabilities isshown in FIG. 13 where the color intensity corresponds to theprobability.

${P\left( {T_{d},\ T_{s}\ ,\ \kappa} \right)} = {\frac{1}{N}{\sum\limits_{m = 1}^{N}{P^{m}\left( {T_{d},T_{s}\ ,\kappa} \right)}}}$

It is noted that in practice it is extremely difficult to obtain scanneddata for arbitrarily varying substrate curvatures (i.e. previouslysolidified droplets). The measurement of the surface curvature has to bedone at the droplet length scale, which may be feasible with the 20 μmresolution scanner, but it is extremely challenging to ensure thedroplet is deposited with such precision that it lands on the previouslysolidified droplet at a region of expected curvature. Therefore, κ=0 isfixed (i.e. a flat substrate) to (2) obtain P(T_(d), T_(s), 0).

Thus, a variation of the classical image restoration problem isobserved; a nominal shape N(T_(d), T_(s), κ) of the droplet can becomputed using a multi-physics PDE solver and an averaged (‘’blurred’)estimate of the printed droplet can be experimentally obtained for afixed set of process parameters. Assuming the uncertainty in dropletdeposition is modeled as a probability kernel D:

P(T _(d) , T _(s), 0)=D(T _(d) , T _(s)κ)⊗N(T _(d) , T _(s), κ)   (3)

Observe that the kernel D(T_(d), T_(s), κ) approximates the probabilityof droplet solidification on a curved surface using data gathered fordroplet solidification on a flat surface; therefore the kernelparameters for non-flat substrates will always be less accurate than theparameters for flat surfaces.

The error D(T_(d), T_(s), κ) in local material deposition (the analog ofthe PSF in image restoration) for a given set of process parameters canbe computed using the classical iterative Richardson-Lucy (RL)deconvolution algorithm.

Algorithm 1 Richardson-Lucy deconvolution Require:

,

 Initialize:

⁽⁰⁾  Define:

 = RotateLeft [

 {N_(i)/2, N_(j)/2}]  repeat   $\mathcal{K}^{({i + 1})} = {\mathcal{K}^{(i)}{\text{?}\left\lbrack {{\text{?}\left\lbrack \frac{\text{?}}{\text{?}\left\lbrack {{\text{?}\left\lbrack {\mathcal{K}\text{?}} \right\rbrack}{\text{?}\left\lbrack \mathcal{T}_{r} \right\rbrack}} \right\rbrack} \right\rbrack}{\text{?}\left\lbrack \mathcal{T}_{r} \right\rbrack}} \right\rbrack}}$ until Σ Flatten [|(

^((i+1)) -

^((i)) |

 ^((i))] < δ Ensure:

indicates data missing or illegible when filed

The output of the RL deconvolution algorithm is an unparameterizedprobability kernel sampled on a grid size equal to the PDE solverresolution. The manufacturing uncertainty may be represented using asmall number of parameters; the 1283 parameters in the binary volumerepresenting the droplet shape are fitted onto a multivariate Gaussianfunction,

$\begin{matrix}{{G\left( {x,y,{z;{\sigma_{x}^{2}\sigma_{y}^{2}\sigma_{Z}^{2}}}} \right)} = {\frac{1}{\left( {2\pi} \right)^{3/2}\sigma_{\chi}\sigma_{y}\sigma_{Z}}{\exp\left( {{- \frac{1}{2}}\left( {\frac{x^{2}}{\sigma_{x}^{2}} + \frac{y^{2}}{\sigma_{y}^{2}} + \frac{z^{2}}{\sigma_{xz}^{2}}} \right)} \right)}}} & (4)\end{matrix}$

Here, σ_(x), σ_(y), σ_(z) are standard deviations estimated byminimizing the norm between the gaussian function and D(T_(d), T_(s)κ).A map between the process parameters and the kernel parameters ({T_(d),T_(s), κ{σ_(x), σ_(y), σ_(z)) using a fully connected neural networkwith hyperbolic tangent activation functions is established. FIG. 14shows a surrogate model for computations discussed herein using a neuralnet. An RL deconvolution process is performed using scanned 3d data fromprinted droplets 1420 and a multi-physics solver for dropletsolidification 1410. The solver 1410 takes for input T_(d), T_(s), andκ. The uncertainty of droplet solidification is approximated as amultivariate Gaussian 1440. The network has an input layer, five hiddenlayers and an output layer. To train the network, droplet shapes arecomputed by running the multiphysics simulation for a single droplet andfor fixed values of {T_(d), T_(s), κ} The output of each simulation atthe final time step is the solidified droplet represented as abinary-valued (indicator function) on a three-dimensional grid, which isused as the nominal shape N(T_(d), T_(s), The uncertainty of dropletsolidification is approximated as a multivariate Gaussian.). Arepresentation of P(T_(d), T_(s), κ) is then computed using datacollected for the fixed set of process parameters. Equation 3, using theRL-deconvolution, is then solved to estimate D(T_(d), T_(s), κ) which isthen fit using the Gaussian function. The process parameters are thenvaried and the training data generation is repeated for eachinstantiated set of parameters to create a large training set, andincrease the accuracy of the neural-net.

According to embodiments described herein, the most accuraterepresentation of multi-drop coalescence would be obtained whileconsidering simultaneous solidification of overlapping and coolingdroplets. It is possible to model this scenario by including thefrequency and spacing of droplet deposition into the multiphysics model,but this type of model may involve exponentially more compute timedepending on the number of solidifying droplets considered. Therefore,in the interest of generating enough training data for the neuralnetwork within a reasonable time frame, it is assumed that liquiddroplets are always deposited on either previously solidified dropletsor on the flat substrate. However, it is noted that in principle thesame idea presented in this section can be extended to build a surrogatemodel for multiple droplet coalescence while considering depositionfrequency as an additional parameter. In practice the computational timethat is involved to build a training set for such a surrogate model maybe prohibitively large.

Variations between the as-designed and as-manufactured shape occurbecause of the cascading error as the droplet is deposited along a scanpath. The uncertainty of droplet solidification is approximated as amultivariate Gaussian. The mapping between input process parameters Td,Ts, κ and the parameters of the Gaussian function are learned. Therepresentation of uncertainty is used at the droplet scale to estimatethe accumulated uncertainty at the part scale. A collection of parts areprinted corresponding to a nominal design H* while fixing the processparameters Td ,Ts and a scalar probability field H(Td ,Ts) isconstructed whose value at a spatial location indicates the probabilityof material deposited at that location. The formalism of Equation 3 isused again to model uncertainty at the part scale.

H(T _(d) , T _(s))=K(T _(d) , T _(s))⊗M(T _(d) , T _(s))   (5)

The field M(T_(d), T_(s)) represents an approximation of the as-builtgeometry while considering droplet coalescence along the tool path. Tocompute this field, it is observed that in principle one may invoke thesurrogate model recursively by setting κ to be the curvature at thepreviously solidified droplet. Even with the assumption that dropletssolidify only on previously solidified droplets or on the flatsubstrate, recursively invoking the machine learning model toapproximate the shape for millions of droplets will take substantialtime (approximately 16 minutes for a million droplets). In a simpleexperiment the time to compute the as-manufactured shape is estimated byinvoking the neural-network for each droplet, and then taking a levelset of the resulting probability field to represent expected dropletshapes with high likelihood. FIG. 15 shows that the time is roughlylinear with the number of invocations of the neural net. To compute theas-manufactured shape rapidly without invoking a solver for eachdroplet's solidification, the assumption is made that the uncertainty atthe droplet scale is shift-invariant, i.e. the uncertainty is notdependent on the position at (6) which an individual droplet isdeposited. This is a reasonable assumption to make because the materialphase-changes and manufacturing uncertainties are not positiondependent. Notice that the same assumption is made in image restorationto model the point-spread function. With this assumption, it is nowobserved that the field M(T_(d), T_(s)) can be modeled as theconvolution

M(T _(d) , T _(s))=G(σ_(x) ²σ_(y) ²σ_(z) ²)⊗T

Here, G(σ_(x) ²σ_(y) ²σ_(z) ²) represents the droplet scale uncertaintyparameterized as a Gaussian function for a specified set of processparameters T_(d), T_(s), κ. In practice it is assumed that κ=0. Now thefield M(T_(d), T_(s)) can be calculated rapidly using the convolutiontheorem and implementing the convolution in frequency domain. Rapidparallel algorithms to compute Fourier transforms are used to quicklyestimate M(T_(d), T_(s)). Our model for M(Td, Ts) is independent of thescan path T used to build the part although in practice the scan pathcan also be a source of deviation between the as-designed andas-manufactured parts. To include the scan path into the formulation ofM(T_(d),T_(s)) we can (similar to the droplet uncertainty estimation)print several distinct plans (i.e. distinct scan paths) of the samenominal design H*, and average the resulting convolutions

$\begin{matrix}{{M\left( {T_{d},T_{s}} \right)} = {\frac{1}{N}{\sum\limits_{i = 1}^{n}{{G\left( {\sigma_{x}^{2}\sigma_{y}^{2}\sigma_{Z}^{2}} \right)} \otimes T_{i}}}}} & (7)\end{matrix}$

Observing that G(σ_(x) ²σ_(y) ²σ_(z) ²) does not change within the sum,and using the property that convolution distributes over addition:

$\begin{matrix}{{M\left( {T_{d},T_{s}} \right)} = {\frac{1}{N}{{G\left( {\sigma_{x}^{2}\sigma_{y}^{2}\sigma_{Z}^{2}} \right)} \otimes {\sum\limits_{i = 1}^{n}T_{i}}}}} & (8)\end{matrix}$

Thus, the calculation of several convolutions can be reduced into asingle convolution over a sum of scan paths. Given voxel representationsof the scan paths, the sum can be computed pointwise and in parallel.Now, suppose we have a description of H(T_(d),T_(s))

$\begin{matrix}{{H\left( {T_{d},T_{s}} \right)} = {\frac{1}{N}{\sum\limits_{i = 1}^{n}{H^{m}\left( {T_{d},T_{s}} \right)}}}} & (9)\end{matrix}$

obtained by printing several specimens H_(m)(T_(d),T_(s)) with fixedprocess parameters T_(d),T_(s), a representation of the family of partsprinted is represented using the same process plan, as a probabilityfield. Thus, the unknown kernel K (T_(d),T_(s)) can be solved by onceagain using the RL-deconvolution algorithm to estimate part leveluncertainty.

Manufacturing error at the part scale is obtained by deconvolving(averaged) scanned prints of a calibration part with a nominalrepresentation generated by convolving droplet-scale uncertainty withscan path(s) representing plans to print the calibration part.

Once K (Td ,Ts) and G(σ_(x) ²σ_(y) ²σ_(z) ²) are determined, therepresentation of the as-manufactured part for a given scan path T givenprocess parameters T_(d), T_(s) can be written as:

H(T _(d) , T _(s))=K(T_(d) , T _(s))⊗(G(σ_(x) ²σ_(y) ²σ_(z) ²)⊗T)   (10)

By associativity, this is equivalent to

H(T _(d) ,T _(s))=(K(T _(d) , T _(s))⊗G(σ_(x) ²σ_(y) ²σ_(z) ²))⊗T   (11)

Writing K (T_(d), T_(s))⊗G(σ_(x) ²σ_(y) ²σ_(z) ²) as X (T_(d), T_(s))(because the Gaussian function parameters are directly related to T_(d),T_(s)):

H(T _(d) , T _(s))=K(T _(d) , T _(s))⊗T   (12)

Equation 12 is used to model the as-manufactured shape corresponding toa scan path and given process parameters. The convolution theorem isused to implement Equation 16 as

H(T _(d) , T _(s))=F ⁻¹(F(X(T _(d) , T _(s)))·F(T)   (13)

Here, · represents pointwise multiplication. This results in accuraterepresentations of the as-manufactured shapes as shown in FIGS. 16A-16C.

The above-described methods can be implemented on a computer usingwell-known computer processors, memory units, storage devices, computersoftware, and other components. A high-level block diagram of such acomputer is illustrated in FIG. 17. Computer 1700 contains a processor1710, which controls the overall operation of the computer 1700 byexecuting computer program instructions which define such operation. Itis to be understood that the processor 1710 can include any type ofdevice capable of executing instructions. For example, the processor1710 may include one or more of a central processing unit (CPU), agraphical processing unit (GPU), a field-programmable gate array (FPGA),and an application-specific integrated circuit (ASIC). The computerprogram instructions may be stored in a storage device 1720 (e.g.,magnetic disk) and loaded into memory 1730 when execution of thecomputer program instructions is desired. Thus, the steps of the methodsdescribed herein may be defined by the computer program instructionsstored in the memory 1730 and controlled by the processor 1710 executingthe computer program instructions. The computer 1700 may include one ormore network interfaces 1750 for communicating with other devices via anetwork. The computer 1700 also includes a user interface 1760 thatenables user interaction with the computer 1700. The user interface 1760may include I/O devices 1762 (e.g., keyboard, mouse, speakers, buttons,etc.) to allow the user to interact with the computer. Such input/outputdevices 1762 may be used in conjunction with a set of computer programsin accordance with embodiments described herein. The user interface alsoincludes a display 1764. According to various embodiments, FIG. 17 is ahigh-level representation of possible components of a computer forillustrative purposes and the computer may contain other components.

Unless otherwise indicated, all numbers expressing feature sizes,amounts, and physical properties used in the specification and claimsare to be understood as being modified in all instances by the term“about.” Accordingly, unless indicated to the contrary, the numericalparameters set forth in the foregoing specification and attached claimsare approximations that can vary depending upon the desired propertiessought to be obtained by those skilled in the art utilizing theteachings disclosed herein. The use of numerical ranges by endpointsincludes all numbers within that range (e.g. 1 to 5 includes 1 1.5, 2,2.75, 3, 3.80, 4, and 5) and any range within that range.

The various embodiments described above may be implemented usingcircuitry and/or software modules that interact to provide particularresults. One of skill in the computing arts can readily implement suchdescribed functionality, either at a modular level or as a whole, usingknowledge generally known in the art. For example, the flowchartsillustrated herein may be used to create computer-readableinstructions/code for execution by a processor. Such instructions may bestored on a computer-readable medium and transferred to the processorfor execution as is known in the art. The structures and proceduresshown above are only a representative example of embodiments that can beused to facilitate embodiments described above. The foregoingdescription of the example embodiments have been presented for thepurposes of illustration and description. It is not intended to beexhaustive or to limit the inventive concepts to the precise formdisclosed. Many modifications and variations are possible in light ofthe above teachings. Any or all features of the disclosed embodimentscan be applied individually or in any combination, not meant to belimiting but purely illustrative. It is intended that the scope belimited by the claims appended herein and not with the detaileddescription.

What is claimed is:
 1. A method comprising: receiving a geometry of asubstrate surface at a neural network, the neural network trained usingone or more training sets, each training set comprising a different typeof substrate geometry, and a collection of manufacturing processparameters, the substrate configured to receive at least one liquiddroplet; determining a shape of the at least one droplet after it hasbeen deposited on the substrate based on the received geometry; andproducing an output representing the determined shape of the at leastone droplet.
 2. The method of claim 1, wherein the shape comprises a 3Drepresentation of the at least one droplet after it has deposited on thesubstrate.
 3. The method of claim 1, wherein the one or more trainingsets comprise a surfaces of varying curvature, comprising one or more ofa smooth surface, a rough surface, and a step-like surface.
 4. Themethod of claim 1, wherein the received geometry of the substratecomprises a 3D representation.
 5. The method of claim 1, wherein theoutput is a 3D representation of the shape obtained by determining theshape based on the received geometry.
 6. The method of claim 1, whereindetermining the shape of the at least one droplet after it has beendeposited on the substrate comprises determining the shape of the atleast one droplet using one or more of a high-fidelity model and asteady-state model.
 7. The method of claim 1, further comprisingestimating a shape of a product part based on the output, the productpart comprising a plurality of droplets.
 8. A method comprising:receiving one or more training sets at a neural network, each trainingset comprising a different type of substrate surface geometry, and acollection of manufacturing process parameters; training the neuralnetwork using the one or more training sets; receiving a geometry of asubstrate at the neural network, the substrate configured to receive atleast one liquid droplet; determining a shape of the at least onedroplet after it has been deposited on the substrate based on thereceived geometry; and producing an output representing the determinedshape of the at least one droplet.
 9. The method of claim 8, wherein theshape comprises a 3D representation of the at least one droplet after ithas deposited on the substrate.
 10. The method of claim 8, wherein theone or more training sets comprise surfaces of varying curvature,comprising one or more of a curved surface, a highly curved surface, arough surface, and a step-like surface.
 11. The method of claim 8,wherein the received geometry of the substrate comprises a 3Drepresentation.
 12. The method of claim 8, wherein the output is a 3Drepresentation of the shape obtained by determining the shape based onthe received geometry.
 13. The method of claim 8, wherein determiningthe shape of the at least one droplet after it has been deposited on thesubstrate comprises determining the shape of the at least one dropletusing one or more of a high-fidelity model and a steady-state model. 14.The method of claim 8, further comprising estimating a shape of aproduct part based on the output, the product part comprising aplurality of droplets.
 15. A system, comprising: a processor; and amemory storing computer program instructions which when executed by theprocessor cause the processor to perform operations comprising:receiving a geometry of a substrate surface at a neural network, theneural network trained using one or more training sets, each trainingset comprising a different type of substrate geometry, and a collectionof manufacturing process parameters, the substrate configured to receiveat least one liquid droplet; determining a shape of the at least onedroplet after it has been deposited on the substrate based on thereceived geometry; and producing an output representing the determinedshape of the at least one droplet.
 16. The system of claim 15, whereinthe shape comprises a 3D representation of the at least one dropletafter it has deposited on the substrate.
 17. The method of claim 15,wherein the one or more training sets comprise surfaces of varyingcurvature, comprising one or more of a curved surface, a highly curvedsurface, a rough surface, and a step-like surface.
 18. The method ofclaim 15, wherein the received geometry of the substrate comprises a 3Drepresentation.
 19. The method of claim 15, wherein the output is a 3Drepresentation of the shape obtained by determining the shape based onthe received geometry.
 20. The method of claim 15, further comprisingestimating a shape of a product part based on the output, the productpart comprising a plurality of droplets.